I don't understand how this DOESN'T contradict Darboux. There is discontinuity in the f'(x) graph because it jumps from -1 to 1. Also it violates the IVT because there is no u for which f(a)<u<f(b).

Please help

As tonio said Darboux's theorem says that if is continuous on and differentiable on then for any there exists some such that . Thus, Darboux's theorem is not violated here since is not differentiable on ANY open ball of . So there is no way to even apply it.